Bi-directional evolutionary topology optimization of geometrically nonlinear continuum structures with stress constraints

This paper proposes a design method to maximize the stiffness of geometrically nonlinear continuum structures subject to volume fraction and maximum von Mises stress constraints. An extended bi-directional evolutionary structural optimization (BESO) method is adopted in this paper. BESO method based on discrete variables can effectively avoid the well-known singularity problem in density-based methods with low density elements. The maximum von Mises stress is approximated by the p-norm global stress. By introducing one Lagrange multiplier, the objective of the traditional stiffness design is augmented with p-norm stress. The stiffness and p-norm stress are considered simultaneously by the Lagrange multiplier method. A heuristic method for determining the Lagrange multiplier is proposed in order to effectively constrain the structural maximum von Mises stress. The sensitivity information for designing variable updates is derived in detail by adjoint method. As for the highly nonlinear stress behavior, the updated scheme takes advantages from two filters respectively of the sensitivity and topology variables to improve convergence. Moreover, the filtered sensitivity numbers are combined with their historical sensitivity information to further stabilize the optimization process. The effectiveness of the proposed method is demonstrated by several benchmark design problems.     

On convergence of extremes under power normalization

In this note, we discuss two aspects of convergence of extremes under power normalization: convergence of moments and convergence of densities. The moments convergence is established for four p-max-stable laws according to conditions imposed on the considered distributions or on the parameter of the p-max-stable laws. For densities convergence, local uniform convergence of the densities is shown to coincide with some von Mises conditions.     

A constructive hypothesis test for the single-index models with two groups

Comparison of two-sample heteroscedastic single-index models, where both the scale and location functions are modeled as single-index models, is studied in this paper. We propose a test for checking the equality of single-index parameters when dimensions of covariates of the two samples are equal. Further, we propose two test statistics based on Kolmogorov–Smirnov and Cramér–von Mises type functionals. These statistics evaluate the difference of the empirical residual processes to test the equality of mean functions of two single-index models. Asymptotic distributions of estimators and test statistics are derived. The Kolmogorov–Smirnov and Cramér–von Mises test statistics can detect local alternatives that converge to the null hypothesis at a parametric convergence rate. To calculate the critical values of Kolmogorov–Smirnov and Cramér–von Mises test statistics, a bootstrap procedure is proposed. Simulation studies and an empirical study demonstrate the performance of the proposed procedures.     

Strong Domain of Attraction of Extreme Generalized Order Statistics

It is a well-known result in extreme value theory that the von Mises conditions imply the strong convergence of extreme order statistics. We extend this result to extreme generalized order statistics. A characterization of strong domains of attraction of joint distributions of a fixed number of extreme generalized order statistics by means of the corresponding result for generalized maxima is given. In particular, we determine the asymptotic joint distribution of (upper and lower) extreme generalized order statistics. Finally, we show that the Hill estimator based on extreme generalized order statistics is asymptotic normal.     

Classical solutions to phase transition problems are smooth

We prove that classical C¹–solutions to phase transition problems, which include the two–phase Stefan problem, are smooth. The problem is reduced to a fixed domain using von Mises variables. The estimates are obtained by frozen coefficients and new L_p estimates for linear parabolic equations with dynamic boundary condition. Crucial ingredients are the observation that a certain function is a Fourier multiplier, an approximation procedure of norms in Besov spaces and Meyer' approach to Nemytakij operators.     

The von Bahr–Esseen moment inequality for pairwise independent random variables and applications

In the paper, we generalize the von Bahr–Esseen moment inequality from independent random variables to pairwise independent random variables. As the applications, the moment convergence, the complete convergence and the strong law of large numbers are established for pairwise independent random variables.     

Nonparametric Estimation of the Measurement Error Model Using Multiple Indicators

This paper considers the nonparametric estimation of the densities of the latent variable and the error term in the standard measurement error model when two or more measurements are available. Using an identification result due to Kotlarski we propose a two-step nonparametric procedure for estimating both densities based on their empirical characteristic functions. We distinguish four cases according to whether the underlying characteristic functions are ordinary smooth or supersmooth. Using the loglog Law and von Mises differentials we show that our nonparametric density estimators are uniformly convergent. We also characterize the rate of uniform convergence in each of the four cases.     

Richard von Mises and the “problem of two races”: A statistical satire in 1934

The article “Problem of two races,” which appeared in 1934 in French and Russian in the Matematichesky Sbornik (Moscow), authored by the noted applied mathematician and statistician Richard von Mises (1883–1953), is interpreted as having a twofold agenda. On its surface it is merely a detailed mathematical analysis of the statistical problem of comparing the distributions of a quantitative characteristic in two different classes. But on closer examination it serves two auxiliary purposes. On the one hand, von Mises, who as a Jew fled Nazi Germany in 1933, satirically attacked Nazi racial doctrines by resorting to statistical parody. On the other hand, von Mises employed his theoretical results to further his own distinctive program of objective Bayesian statistics.     

Plastic flow in a conical channel: Qualitative features of the solutions under different yield conditions

The problem of the convergence of the solutions of problems of plasticity theory, with a yield condition which depends on the hydrostatic stress, to solutions based on classical plasticity theory with von Mises or Tresea conditions is considered, with a particular choice of the parameters of the material model. For the case of axisymmetric flow of material in a channel with converging and diverging walls, solutions according to two plasticity theories with a yield condition which depends on the hydrostatic stress are compared with the classical solution. It is shown that only the solution using Spencer's model shows all the main features of the classical solution. As the internal criterion of the choice of the preferred plasticity theory when examining a special class of problems, it is suggested that the criterion of the convergence of the solutions to the solutions of classical plasticity theory should be used.     

Estimating the inter-arrival time density of Markov renewal processes under structural assumptions on the transition distribution

We consider a stationary Markov renewal process whose inter-arrival time density depends multiplicatively on the distance between the past and present state of the embedded chain. This is appropriate when the jump size is governed by influences that accumulate over time. Then we can construct an estimator for the inter-arrival time density that has the parametric rate of convergence. The estimator is a local von Mises statistic. The result carries over to the corresponding semi-Markov process.     

Probabilistic analysis of a differential equation for linear programming

In this paper we address the complexity of solving linear programming problems with a set of differential equations that converge to a fixed point that represents the optimal solution. Assuming a probabilistic model, where the inputs are i.i.d. Gaussian variables, we compute the distribution of the convergence rate to the attracting fixed point. Using the framework of Random Matrix Theory, we derive a simple expression for this distribution in the asymptotic limit of large problem size. In this limit, we find the surprising result that the distribution of the convergence rate is a scaling function of a single variable. This scaling variable combines the convergence rate with the problem size (i.e., the number of variables and the number of constraints). We also estimate numerically the distribution of the computation time to an approximate solution, which is the time required to reach a vicinity of the attracting fixed point. We find that it is also a scaling function. Using the problem size dependence of the distribution functions, we derive high probability bounds on the convergence rates and on the computation times to the approximate solution.     

A multivariate empirical characteristic function test of independence with normal marginals

This paper proposes a semi-parametric test of independence (or serial independence) between marginal vectors each of which is normally distributed but without assuming the joint normality of these marginal vectors. The test statistic is a Cramér–von Mises functional of a process defined from the empirical characteristic function. This process is defined similarly as the process of Ghoudi et al. [J. Multivariate Anal. 79 (2001) 191] built from the empirical distribution function and used to test for independence between univariate marginal variables. The test statistic can be represented as a V-statistic. It is consistent to detect any form of dependence. The weak convergence of the process is derived. The asymptotic distribution of the Cramér–von Mises functionals is approximated by the Cornish–Fisher expansion using a recursive formula for cumulants and inversion of the characteristic function with numerical evaluation of the eigenvalues. The test statistic is finally compared with Wilks statistic for testing the parametric hypothesis of independence in the one-way MANOVA model with random effects.     

The steady axisymmetric flow of ideally plastic materials in a conical channel

The problem of the asymmetric flow of an ideally plastic medium is formulated within the framework of the von Mises model and the total plasticity condition, using the invariant condition of compatibility for the deviator component of the stress tensor. Flow in a converging conical channel, on the boundary of which the shear stresses are specified, is considered. First-order differential equations are obtained, describing the shear-stress distribution in the moving medium, one of which corresponds to the von Mises model, and the other to the total plasticity condition. It is established from an analysis of the solution in the neighbourhood of singular points, that the minus sign in front of the radical in these equations corresponds to positive shear stresses and vice versa. The problem of the shear stresses reaching a maximum value on the specified boundary surface of the channel is investigated. The aperture angle of the channel, beginning from which this value is reached, is determined. It is established that the value of the angle, following from the total plasticity condition, somewhat exceeds its value obtained within the framework of the von Mises model.     

The solution of the axisymmetric elastic-plastic torsion of a shaft using varational inequalities

The axisymmetric elastic-plastic torsion of a shaft subject to the von Mises yield criterion is considered. The problem is reformulated as a variational inequality and it is proved that the problem has a unique solution. Some properties of the solution are derived.     

An efficient simulation algorithm for the generalized von Mises distribution of order two

In this article we propose an exact efficient simulation algorithm for the generalized von Mises circular distribution of order two. It is an acceptance-rejection algorithm with a piecewise linear envelope based on the local extrema and the inflexion points of the generalized von Mises density of order two. We show that these points can be obtained from the roots of polynomials and degrees four and eight, which can be easily obtained by the methods of Ferrari and Weierstrass. A comparative study with the von Neumann acceptance-rejection, with the ratio-of-uniforms and with a Markov chain Monte Carlo algorithms shows that this new method is generally the most efficient.     

Convergence rates for weighted sums in noncommutative probability space

We study convergence rates for weighted sums of pairwise independent random variables in a noncommutative probability space of which the weights are in a von Neumann algebra. As applications, we first study convergence rates for weighted sums of random variables in the noncommutative Lorentz space, and second we study convergence rates for weighted sums of probability measures with respect to the free additive convolution.     

Tests of independence among continuous random vectors based on Cramér–von Mises functionals of the empirical copula process

A decomposition of the independence empirical copula process into a finite number of asymptotically independent sub-processes was studied by Deheuvels. Starting from this decomposition, Genest and Rémillard recently investigated tests of independence among random variables based on Cramér–von Mises statistics derived from the sub-processes. A generalization of Deheuvels’ decomposition to the case where independence is to be tested among continuous random vectors is presented. The asymptotic behavior of the resulting collection of Cramér–von Mises statistics is derived. It is shown that they are not distribution-free. One way of carrying out the resulting tests of independence then involves using the bootstrap or the permutation methodology. The former is shown to behave consistently, while the latter is employed in practice. Finally, simulations are used to study the finite-sample behavior of the tests.     

Reliable solution of an elasto-plastic Reissner-Mindlin beam for Hencky's model with uncertain yield function

We apply the method of reliable solutions to the bending problem for an elasto-plastic beam, considering the yield function of the von Mises type with uncertain coefficients. The compatibility method is used to find the moments and shear forces. Then we solve a maximization problem for these quantities with respect to the uncertain input data.     

Convergence analysis of sample average approximation for a class of stochastic nonlinear complementarity problems: from two-stage to multistage

In this paper, we consider the sample average approximation (SAA) approach for a class of stochastic nonlinear complementarity problems (SNCPs) and study the corresponding convergence properties. We first investigate the convergence of the SAA counterparts of two-stage SNCPs when the first-stage problem is continuously differentiable and the second-stage problem is locally Lipschitz continuous. After that, we extend the convergence results to a class of multistage SNCPs whose decision variable of each stage is influenced only by the decision variables of adjacent stages. Finally, some preliminary numerical tests are presented to illustrate the convergence results.

     

理想塑性轴对称问题的一般方程

本文引用速度势函数,将理想塑性轴对称问题化为二个非线性的偏微分方程,根据导得的方程讨论了Haar-Kármán假设对于Mises屈服准则及与其相关联的流动法则的协调性问题。